Distribution of Mutations

Biostatistics 666Lecture 5

Last Lecture:Introduction to the Coalescent

Coalescent approach• Proceed backwards through time.• Genealogy of a sample of sequences.

Infinite sites model• All mutations distinguishable.• No reverse mutation.

Some key ideas …

Probability of coalescence events

Length of genealogy and its branches

Expected number of mutations

Parameter θ which combines population size and mutation rate

Building Blocks…

Probability of sampling distinct ancestors for n sequences

Coalescence time t is approximately exponentially distributed

N

n

NinP

n

i

⎟⎟⎠

⎞⎜⎜⎝

⎛

−≈⎟⎠⎞

⎜⎝⎛ −=∏

−

=

211)(

1

1

Some Key Results…

Coalescence Time (population size units)

Total Length (population size units)

∑−

=

=1

1

2)(n

itot i

TE

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2/1)(

jTE j

Some More Key Results …

Expected Number of Polymorphisms

∑∑

∑∑

−

=

−

=

−

=

−

=

==

==

1

1

1

1

1

1

1

1

/1/12)(

sample haploidan For

/1/14)(

sample diploid aFor

n

i

n

i

n

i

n

i

iiNSE

iiNSE

θµ

θµ

Inferences about θ

Could be estimated from S• Divide by expected length of genealogy

Could then be used to:• Estimate N, if mutation rate µ is known• Estimate µ, if population size N is known

∑−

=

= 1

1/1

ˆn

ii

Sθ

Alternative Estimator for θ …

Count pairwise differences between sequences

Compute average number of differences

∑∑= +=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

i

n

ijijS

n

1 1

1

2~θ

Var(θ) as a function of N

2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50

Sample Size

Var

ianc

e in

Est

imat

e of

The

ta

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Parameters

N = 10,000 individualsµ = 10-4

θ = 4

^

Today ...

More applications of the coalescent

Predicting allele frequency distributions• Using simulations

The full distribution of S• Using analytical calculations

A Coalescent Simulation …

Let’s consider tracing the ancestry of 4 sequences

When n = 4

coalesce torandomat sequences Pick twoondistributi lexponentia from timeSample

24

2)4(

Event CoalescentNext toTime

224

)4(

Event Coalescent ofy Probabilit

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

NT

NP

Next n = 3 …

T(4)

Let’s assume that sequences 3 and 4 are selected …

Then, we repeat the process for a sample of 3 sequences

Next n = 2 …

T(4)

Let’s assume that sequences 1 and 2 are selected to coalesce

Then, we repeat the process for a sample of 2 sequences

T(3)

The Simulated Coalescent

T(4)

T(3)

T(2)At this point, we could

place mutations in genealogy. Most often,

these would fall in longer branches.

A Coalescent Simulation …

T(4)

T(3)

T(2)Mutations in these

branches affecta pair of sequences

A Coalescent Simulation …

T(4)

T(3)

T(2)Mutations in these

branches affecta single sequence

Frequency Spectrum

Repeating the simulation multiple times, would give us a predicted mutation spectrum.

Frequency (out of n)

Pro

babi

lity

0.0

0.1

0.2

0.3

0.4

0.5

Frequency Spectrum (n = 10)

1 2 3 4 5 6 7 8 9 10

Frequency (out of n)

Pro

porti

on o

f All

Var

iant

s

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Frequency Spectrum (n = 100)

1 4 7 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Frequency (out of n)

Pro

porti

on o

f All

Var

iant

s

0.00

0.05

0.10

0.15

Frequency Spectrum

Constant size populationExponentially growing population

Most variants are rare• For n = 100, ~44% of variants occur < 5/100.• For n = 10, ~35% of variants observed once.

Mutation Spectrum

Depends on genealogy• Population Size• Population Growth• Population Subdivision

Does not depend on• Mutation rate!

Deviations from Neutral Spectrum

When would you expect deviations from the spectra we described?

What would you expect for …• A rapidly growing population?• A population whose size is decreasing?

Why?

Effect of Polymorphism Type

Data from Cargill et al, 1999

Number of Mutations

Can be derived from coalescent tree• What are the key features?

Analytical results possible• Trace back in time until MRCA, tracking

mutation events

Sample of Two Sequences

Track coalescences and mutations• Probability of a coalescent event?

• Depends on population size …• Probability of a mutation?

• Depends on mutation rate …

Proceed backwards until either occurs…• Conditional probability for each outcome?

Two Identical Sequences

θ

µ

+=

+=

+≈

11

22/12/1

)0 is (2

NN

PPPSP

mutCA

CA

Full distribution of S…

Probability that first j events are mutations…

⎟⎠⎞

⎜⎝⎛

+⎟⎠⎞

⎜⎝⎛

+=

θθθ

11

1)(2

j

jP

Example…

2 sequencesPopulation size N = 25,000Mutation rate µ = 10-5

Probability of 0, 1, 2, 3… mutations

And for multiple sequences…

Describe number of mutations until the next coalescence event

Proceed back in time, until:• One of n sequences mutates…• A coalescent event occurs…

• Then track mutations in (n-1) sequences

Formulae …

11

1

22

22

22

)(−+

−⎟⎠⎞

⎜⎝⎛

−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛

+

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+

=n

nn

N

n

n

N

n

N

n

n

njQj

j

n θθθ

µµ

µ

∑=

− −=j

innn iQijPjP

01 )()()(

Example…

3 sequencesPopulation size N = 25,000Mutation rate µ = 10-5

Probability of 0, 1, 2, 3… mutations

Number of Mutations

0 1 2 3 4 5 6 7 8 9 10+

Number of Mutations

Pro

babi

lity

0.00

0.05

0.10

0.15

0.20

N = 10,000n = 10µ = 2 * 10-5

Approximately 2kb of sequence,sequenced in 10 individuals

So far …

One homogeneous population• Coalescence times• Number of mutations

• Expectation• Distribution

• Spectrum of mutations

Several assumptions, including …• No recombination

Recombination …

No recombination• Single genealogy

Free recombination• Two independent genealogies• Same population history

Intermediate case• Correlated genealogies

Recommended Reading

Richard R. Hudson (1990) Gene genealogies and the coalescent process

Oxford Surveys in Evolutionary Biology, Vol. 7.D. Futuyma and J. Antonovics (Eds). Oxford University Press, New York.